INSTITUTE OF PHYSICS PUBLISHING
Physiol. Meas. 26 (2005) 59–67
PHYSIOLOGICAL MEASUREMENT
doi:10.1088/0967-3334/26/1/006
Measurement of temperature-dependent specific heat
of biological tissues
Dieter Haemmerich1, David J Schutt2, Icaro dos Santos3,
John G Webster2 and David M Mahvi4
1 Division of Pediatric Cardiology, Medical University of South Carolina, 165 Ashley Ave,
Charleston, SC 29425, USA
2 Department of Biomedical Engineering, University of Wisconsin-Madison,
1550 Engineering Drive, Madison, WI 53706, USA
3 Department of Electrical Engineering, University of Brasilia, Brasilia DF, 70..919-970, Brazil
4 Department of Surgery, University of Wisconsin-Madison, 600 Highland Avenue, Madison,
WI 53792, USA
E-mail: [email protected]
Received 8 July 2004, accepted for publication 1 December 2004
Published 10 January 2005
Online at stacks.iop.org/PM/26/59
Abstract
We measured specific heat directly by heating a sample uniformly between
two electrodes by an electric generator. We minimized heat loss by
styrofoam insulation. We measured temperature from multiple thermocouples
at temperatures from 25 ◦ C to 80 ◦ C while heating the sample, and corrected for
heat loss. We confirm method accuracy with a 2.5% agar–0.4% saline physical
model and obtain specific heat of 4121 ± 89 J (kg K)−1, with an average error
of 3.1%.
Keywords: specific heat, measurement of specific heat, tissue specific heat
measurement
1. Introduction
This paper presents a method to measure temperature-dependent specific heat of tissues.
Knowledge of temperature-dependent specific heat is required when investigating thermal
therapies (e.g., thermal tumor treatment) and other biological heat transfer problems. Data on
specific heat of different tissue types are available (Duck 1990, Rahman 1995), though few
studies evaluate the specific heat at temperatures above 50 ◦ C. A number of different methods
for measuring specific heat are available (Touloukian and Buyco 1970). Differential scanning
calorimetry (DSC) is the most practical tool to measure temperature-dependent specific heat of
biological materials (Mohesnin 1980, Sweat 1986). DSC requires small samples (millimeter
thickness), since uniform sample temperature must be guaranteed. In small samples however,
0967-3334/05/010059+09$30.00 © 2005 IOP Publishing Ltd Printed in the UK
59
60
D Haemmerich et al
Electrode
Thermocouples
P
Generator
Sample
Styrofoam
Figure 1. Experimental set-up (side view): A sample (5 × 5 cm2, 4 cm thick) is placed between
two metal electrodes. The sample is heated with constant power supplied by a generator, and
thermally insulated by 1 cm thick expanded polystyrene. Sample temperature is monitored by six
thermocouples.
water bound in the tissue may evaporate markedly below the boiling temperature of water
(Ramachandran et al 1996), depending on how tight the sealing of the sample is. A previous
study reports an increase of heat flow during DSC of liver tissue (which is directly correlated to
specific heat) of ∼5 times at 80 ◦ C compared to values below 70 ◦ C due to water vaporization
from the sample (Ramachandran et al 1996). Different seal conditions produce different
results when DSC is used and especially at higher temperatures errors can be introduced
(Tang et al 1991). It may therefore be difficult to choose the seal condition that corresponds
to an in vivo setting where tissue is heated inside a solid organ.
We present a simple method that allows measurement of specific heat of large tissue
samples up to boiling temperatures, and verify the method in phantom measurements.
2. Theory
The heat transfer equation when a sample with heat capacity c is heated by applying power
with a mass-specific energy rate p (W kg−1) is
∂T
∇ · k∇T
c
=
+ p.
(1)
∂t
ρ
To measure temperature-dependent specific heat c, the temperature distribution inside the
sample must be known so that the term describing thermal conduction (first term on righthand side) is known. Furthermore, power density at a location where temperature is measured
must be known.
We attempted to obtain uniform power density in a large sample with the set-up shown
in figure 1. A cube-shaped sample with known dimensions and mass is placed between two
plate electrodes. Under ideal conditions, this set-up produces uniform heating of the sample.
The sample is thermally isolated so that no heat is lost, resulting in uniform temperature
distribution in the sample, i.e., the thermal conduction term in equation (1) is zero. Under
these conditions, we can obtain the specific heat of the sample by measuring the temperature
at an arbitrary location:
P
c=
(2)
m · (∂T /∂t)
where P is the total power applied to the sample and m is the sample mass.
Measurement of temperature-dependent specific heat of biological tissues
61
T
Slope at t +off
Stop power
application at t off
t
Figure 2. Typical temperature–time course. Power application is stopped at toff . Slope after power
shutdown is used to correct for error resulting from conductive losses.
The following assumptions are made:
•
•
•
•
homogeneous sample,
negligible heat loss to the environment,
uniform contact between electrodes and sample surfaces,
sample mass does not change (e.g., due to vaporization).
2.1. Correction for heat loss
From preliminary experiments, we found that at high temperatures (>50 ◦ C) the heat
conduction term was not negligible in our set-up; we observed a significant decline in
temperature after power was turned off. One way to reduce this error would be to increase
the width of the sample so that the temperature profile near the center is more homogeneous.
However, we can also correct for this error using the following method. Figure 2 shows the
temperature–time course during an experiment. The temperature rise during power application
is almost linear, followed by a slow exponential decay after power is shut off at toff . Since
the temperature distribution is the same right before and after the power is turned off, the heat
conduction term (see equation (1)) also has to be identical. The heat transfer equation right
−
) is
before toff (i.e., toff
∂T ∇ · k∇T
c
=
+ p.
(3)
∂t −
ρ
toff
Right after toff , no power is applied and equation (3) becomes
∂T ∇ · k∇T
c
=
.
∂t toff
ρ
+
(4)
In equations (3) and (4), the time derivatives of temperature ∂T/∂t right before and after toff
are the slopes of the temperature–time course (see figure 2), which we can obtain from the
experiments. If the temperature decline after power shut down is significant, we can use
equations (3) and (4) to correct for thermal conduction-related temperature changes; from
these equations, we can calculate the specific heat according to
P
.
∂T c=
(5)
m · ∂t t − − ∂T
∂t t +
off
off
62
D Haemmerich et al
This equation is very similar to equation (2), except we introduced a correction term—the
slope of the temperature decrease after power shut down. We can only obtain this correction
term at temperatures where we turn the power off, so we can measure the slope of temperature
decrease, i.e., we can only obtain the correction term at a number of distinct temperatures.
To apply the correction for other temperatures within the measurement temperature range, we
can interpolate the correction term between the distinct temperatures where it was measured.
The experimental procedure to determine specific heat is then:
• heat to temperature T1, and turn off power to determine slope of temperature decline
• heat to temperature T2, and turn off power to determine slope of temperature decline,
and so on.
Thereby, we are able to directly measure specific heat of the sample, without need for
calibration with a substance of known specific heat.
3. Methods
3.1. Numerical analysis
We used finite element method analysis to determine temperature distribution in a sample
(5 × 5 × 4 cm3) insulated by a 1 cm layer of expanded polystyrene (see figure 1). Thermal
and electrical properties of 0.4% saline (i.e., the properties of the phantom used in the
experiments) were used for the sample. A constant power of 60 W was applied to the
sample by application of a voltage difference across the sample (top to bottom in figure 1),
until the temperature at the center reached 80 ◦ C. A temperature of 25 ◦ C was applied to the
outer boundary of the model.
All simulations were performed using ABAQUS finite element software on a Sun Blade
1000 workstation with 2 GB of memory and 80 GB of hard disk space. The 3D model had
∼10 000 elements. Haemmerich et al (2002) describe the modeling procedure in more detail.
3.2. Experimental set-up
Figure 1 shows the experimental set-up. Our samples were made of 2.5% agar–water (mass
fraction), and were prepared using a 0.4% saline solution (mass fraction) to provide an
electrically conductive sample that can be heated by application of electric current. The specific
heat of this sample is the same as that of water, and changes little with temperature; the salt
only causes a slight change in specific heat compared to water; the specific heat of our sample
is c = 4160 J (kg K)−1 at 25 ◦ C, with a slight increase to 4178 J (kg K)−1 at 80 ◦ C (Incropera
and DeWitt 1996). The sample was cut from an agar–water block with a surgical blade (5 ×
5 × 4 cm3, measured with a standard ruler). The sample was placed between two plate
electrodes (5.0 × 5.0 cm2) made of sheet metal (Zn-plated Fe). A thin layer of conductive
gel was applied to the electrodes to reduce errors from uneven sample surfaces that may
result in uneven contact. The sample was thermally insulated by 1 cm thick expanded
polystyrene (k = 0.03 W (m K)−1) on each side. We placed a weight of 200 g on top of
this set-up to ensure electric contact between sample and electrodes. Six electrically insulated
thermocouples (accuracy 0.2 ◦ C) were placed in the center plane between the electrodes inside
the sample as shown in figure 3. The temperature was recorded at 5 samples/s during the
experiment.
A commercial generator was used to heat the sample (PDX-500, Advanced Energy) by
applying electric current of 375 kHz with constant power to the sample. We applied 60 W
(accuracy 2 W) of power to heat the sample, which resulted in a heating rate of ∼10 ◦ C min−1.
Measurement of temperature-dependent specific heat of biological tissues
63
5 cm
2.5 cm
5 cm
1 cm
2.5 cm
1 cm
Figure 3. Sample (top view, center plane), with six thermocouple locations.
90
80
70
T (°C)
60
50
40
30
20
10
0
0
100
200
300
400
500
600
700
800
t (s)
Figure 4. Typical temperature–time course of six temperatures measured by thermocouples during
an experiment. Sample was heated to 40, 60, 70 and 80 ◦ C, with intermittent 1.5 min cool-down
cycles.
(This figure is in colour only in the electronic version)
To allow for correction as described above, we had to include cool-down cycles. Starting at
room temperature (25 ◦ C), we heated to 40, 60, 70 and 80 ◦ C, with an intermittent 1.5 min
cool-down cycle after each of the temperatures was reached. Before and after each experiment,
we measured the mass of the sample with a scale (OHaus CS-200, 0.1 g accuracy). Sample
masses were between 85 and 100 g, and were reduced by ∼2 g after the heating, possibly due
to water loss.
3.3. Data evaluation
We calculated the initial slope of temperature decrease after power was turned off at 40, 60,
70 and 80 ◦ C, to correct for errors caused by thermal conduction as described above. We
used a linear approximation from these data points to correct for this error at all temperatures
between 25 ◦ C and 80 ◦ C (see figure 4). We averaged the temperature of the six thermocouples,
excluding up to two thermocouple readings that were outliers (i.e., difference was >5 ◦ C
compared to other temperatures). We determined the slope of temperature rise during heating;
since this derivative is very sensitive to noise in the data, we used moving averages with a
window size of 2 ◦ C to reduce noise. We further measured the initial slope of temperature
64
D Haemmerich et al
80 °C
25°C
Figure 5. Temperature distribution in computer model. A sample (white rectangle) surrounded by
styrofoam was heated to 80 ◦ C by application of a voltage drop from top to bottom surface. The
model confirms uniform heating of the sample.
decay after power shutdown. From these data we calculated the specific heat of the sample
using equation (5) within our measurement temperature range of 25–80 ◦ C. For the sample
mass, we used the average of the mass before and after the heating, since the weight loss was
comparably small, and we did not know at which temperature the mass loss occurred. We
performed a total of eight experiments in four samples. For each experiment, we determined
error versus temperature within the measurement temperature range. We then calculated
average error at each temperature for our eight experiments, by comparing the value to the
known specific heat of the sample.
4. Results and discussion
Figure 4 shows the temperature–time course measured by the six thermocouples during
one of the experiments. There is very little deviation in the temperatures measured at the
different sample locations, confirming uniform heating of the sample. Figure 4 also shows
that temperature decrease during the cool-down cycle is noticeable as higher temperatures are
reached. Figure 5 shows the temperature distribution of the computer model. The temperature
is uniform except at the edges, confirming our assumption of uniform heating. Figure 6 shows
the initial slope of temperature decay after power is turned off for the different temperatures
of 40, 60, 70 and 80 ◦ C; as the slope changes fairly linearly with temperature, we used a
linear approximation to correct at all temperatures between 25 ◦ C and 80 ◦ C in equation (5);
R 2 was between 0.96 and 0.99. Figure 7 shows the temperature-dependent specific heat c
obtained in one of the experiments; results are shown both with correction for the error caused
by thermal conduction using equation (5), and without correction using equation (2). The
deviation between the two curves increases with higher temperatures, as sample cooling due
to thermal conduction losses intensifies with temperature. The same is apparent in figure 8,
which shows the measurement error at different temperatures, averaged over eight samples,
with and without error correction. Figure 9 shows the temperature dependence of specific heat
(Average ± StdDev of eight experiments); the actual specific heat is represented by the dashed
line. The error averaged over eight samples, and averaged over the measurement temperature
range (25–80 ◦ C) was 3.1 ± 1.6%; due to the small change of specific heat of our sample with
temperature (∼0.3% change between 25 ◦ C and 80 ◦ C), averaging between 25 ◦ C and 80 ◦ C
Measurement of temperature-dependent specific heat of biological tissues
65
0.015
dT /dt ( °C/s)
0.01
0.005
0
40
45
50
55
60
65
70
75
80
T ( °C)
Figure 6. Initial slope of temperature decrease at 40, 60, 70 and 80 ◦ C (diamonds), with linear
approximation (line) (experiment #4). R 2 = 0.98.
5000
c (J/kg/K)
4000
3000
2000
1000
0
25
35
45
55
65
75
T (°C)
Figure 7. Measured specific heat with (gray), and without correction (black) (experiment #4). The
actual specific heat of the sample was 4160 W (kg K)−1.
is justified. Table 1 shows the measured values of specific heat for each experiment, including
error averaged over the measurement temperature range.
The following are the sources contributing to the error:
• shape of the sample is not ideal,
• sample mass changes during the experiment,
• non-uniform contact between electrodes and sample due to uneven electrode and sample
surfaces,
• measurement errors in temperature, applied power and sample mass,
• sample inhomogeneity (unlikely a factor with our agar–water sample, but may be
important, e.g., in biological samples).
Temperature measurements at several locations allow reduced errors introduced by sample
inhomogeneities. Larger sample dimensions reduce errors from deviations in sample shape,
and non-uniform electrode–sample contact; the sample temperature can be measured farther
away from electrodes where heating is more uniform than close to the electrodes.
66
D Haemmerich et al
16
14
12
err (%)
10
8
6
4
2
0
25
35
45
55
65
75
T (°C)
Figure 8. Error (average of eight experiments) with (black line, ± StdDev) and without (gray line,
± StdDev) correction.
5000
4500
4000
c (J/kg/K)
3500
3000
2500
2000
1500
1000
500
0
25
35
45
55
65
75
T (°C)
Figure 9. Specific heat (average of eight experiments, with StdDev). The actual specific heat is
indicated by the dashed line.
Table 1. Average specific heat, average error and maximum error, averaged over measurement
temperature range (25–80 ◦ C), for each experiment.
Experiment #
cave (J kg−1 K−1)
Errave (%)
Errmax (%)
1
2
3
4
5
6
7
8
Ave
4279 ± 107
3920 ± 55
4125 ± 105
4284 ± 89
4149 ± 98
4095 ± 91
4140 ± 81
3977 ± 84
4121 ± 89
3.4 ± 2.5
5.8 ± 1.3
2.2 ± 1.5
3.1 ± 2.0
2.0 ± 1.2
2.3 ± 1.4
1.7 ± 1.1
4.5 ± 1.9
3.1 ± 1.6
9.5
10.0
7.7
10.2
6.9
6.5
5.7
8.4
8.1 ± 1.7
Last line shows averages of eight experiments.
Measurement of temperature-dependent specific heat of biological tissues
67
5. Conclusion
For biological materials, DSC is commonly used to determine temperature-dependent specific
heat of biological materials. For DSC small samples are required, where errors may be
introduced especially at high temperatures depending on sample sealing.
We present a simple instrument that allows measurement of temperature-dependent
specific heat of large samples, and confirmed performance by measurements in phantoms.
The average error was 3.1 ± 1.6%, which is sufficient for measurement of biological tissues
where the typical variability is in the range of 5–10%. The error may be larger for measurement
of tissues due to inhomogeneities.
Acknowledgment
This study was supported by the National Institute of Health (NIH) under grant DK58839.
References
Duck F A 1990 Physical Properties of Tissue (London: Academic) pp 167–223
Haemmerich D, Tungjitkusolmun S, Staelin S T, Lee F T Jr, Mahvi D M and Webster J G 2002 Finite-element
analysis of hepatic multiple probe radio-frequency ablation IEEE Trans. Biomed. Eng. 49 836–42
Incropera F P and DeWitt D P 1996 Fundamentals of Heat and Mass Transfer (New York: Wiley) pp 846–7
Mohesnin N M 1980 Thermal Properties of Food and Agricultural Materials (New York: Gordon and Breach)
Rahman S 1995 Food Properties Handbook (Boca Raton, FL: CRC Press)
Ramachandran T, Sreenivasan K and Sivakumar R 1996 Water vaporization from heated tissue: an in vitro study by
differential scanning calorimetry lasers Surg. Med. 19 413–5
Sweat V E 1986 Thermal Properties of Foods (New York: Dekker)
Tang J, Sokhansanj S, Yannacopoulos S and Kasap S O 1991 Specific-heat capacity of lentil seeds by differential
scanning calorimetry Trans. ASAE 34 517–22
Touloukian Y S and Buyco E H 1970 Thermophysical Properties of Matter (New York: IFI/Plenum)